Asymmetry of the quasiparticle distribution in
superconductors and normal metals
A. F. Volkov and A. V. Zaltsev
Institute of Radio Engineering and Electronics, USSR Academy of Sciences
(Submitted July 10, 1975)
Zh. Eksp. Teor. Fiz. 69, 2222-2230 (December 1975)
The population asymmetry of the energy spectrum branches (~> 0 and ~ < 0; ~ = v(p - PF» that is produced
in a superconductor S upon the tunnel injection of nonequilibrium quasiparticJes into S is theoretically
investigated. It is shown that such an asymmetry leads to the appearance in S of a gauge-invariant
potential 4> = If +( 1I2)X and to the appearance of a voltage potential in the measuring circuit. The steady-state
value of the voltage potential is computed. It is also shown that if the electron-hole distribution in a
normal metal N is made asymmetric through injection, then a potential difference arises between Nand
the measuring electrode if as the latter a superconductor Sm... is used.
PACS numbers: 74.30.-e, 71.85.-a
Clarke, Peterson, and Tinkham[l-3J have carried out
experimental and theoretical investigations of the tunnel
structure S'-S-N (Fig. 1). A current I was passed
through the structure in such a way that quasiparticles
were continuously injected from N into S. The neutrality in S was maintained as a result of the drift of
Cooper pairs into S'. It was observed that a voltage
potential tf existed in the measuring circuit whenever
the injecting current I was different from zero. As has
been shown by Clarke and Tinkham, the appearance of
the voltage potential is due to the asymmetry of the excitation distribution function n( ~) in the superconductor
with respect to ~ (~ = v(p - PF)). This means that the
number N> (/; > 0) of particles on the n-type excitation
branch is not equal to the number N < (I; < 0) of particles on the p-type excitation branch. Tinkham computed
the time for the establishment of the steady-state value
of Q = N> - N <, and found that in the case of scattering
of the quasiparticles by phonons this time is TQ ~ .0. -\
since in the normal metal the collision integral leaves
the quantity Q unchanged (in the normal metal Q is the
difference between the number of electrons and the number of holes, and therefore the invariability of Q implies the conservation of the total number of particles).
The time TQ determines the attenuation length of the
longitudinal electric field E in a superconductor with a
nonzero energy gape 4 J. Tinkham also calculated the
quantity iff, and found that tf ~ Q"'TQ, where Q!:::! Q'"
near the critical temperature Tc. He did not, however,
take into account the appearance in S of the gauge-invariant potential
(f/J is the electrical potential and X is the phase of the
order parameter), and the electrical neutrality of the
superconductor S was not properly taken into account.
Meanwhile, as will be shown below, the voltage potential
rff is due precisely to the presence of the potential 4>.
Such a potential arises if the di vergence of the supercurrent is different zero, which is the case when, for
example, current is passed across an S-N junction!) [S,6J.
In a previous paper by one of the present authors[9 J,
the growth rate of 4> was found (in the cp '" 0 gauge)
without allowance for the collision integral. Here we
shall find the steady-state values of 4> and iff, taking
into account the inelastic collisions with phonons.
We shall also consider the tunnel structure N I-N -N2'
and show that the electron and hole distributions in the
1130
SOY.
Phys.·JETP, Vol. 42, No.6
FIG. I. The system under consideration: the hatched regions represent insulator layers, I is the injection current, and ~ is the voltage
poten tial to be measured.
metal N become asymmetric when a current is passed
through such a structure (although the total number of
electrons remains equal to the total number of holes,
i.e., Q = 0). Such an asymmetry leads also to the appearance of a potential difference between N and the
measuring electrode if as the latter a superconductor
is used.
1. THE SYSTEM S'-S-N
For greater physical clarity, we use a computational
method somewhat different from the one used earlier
inC 9J. It is convenient to express the Green functions
with the aid of which the calculation in[9J was carried
out in terms of the occupation number n( ~) of the
quasiparticles, as is done by Aronov and Gurevich in[lOJ.
Let us find the rate of change, due to the injection,
of the number of quasiparticles in S in terms of the
functions G12 and G21 (loJ:
an =_l_!..Soo dUl(G."-G_.
at 2Jli at"
ZI ).
(1 )
Let us write down the equation for G[ 10, 11 J:
(i a~, -s) GI2(tlt2)=~IlG"+L:"G"~A(t"t,),
GI2(t" t,) =A·(t"
( i~+S)
at,
(2 )
t ,),
where :0 is the self-energy part responsible for the
BCS interaction and the tunneling of quasiparticles from
N into S[91. Let us add the Eqs. (2) and carry out a
Fourier transformation with respect to the difference
variable (t 1 - t 2 ). We obtain
i~G."=2 Re A.(t).
at
(3)
Similarly, for G~ we find
-i!..G
ZI=2Re(L: Zl G"+L:"G ZI ) W,
rJt
(4)
(iI
Copyright © 1976 American Institute of Physics
1130
where t = 7'2 (t 1 + t 2). F~rther, let. us substitute the
equilibrium values of ~t~n and Glk into the right-hand
sides of (3) and (4), as was done in[9] in the computation
of the sources JS-N' Using (1), we have
[e+v
e-·V
e
S ( e+V
- = - - t h - - + t h - - - 2 t l l - + - Lll-On
\I
at
2
21'
21'
21'
e
(5)
21'
-th e;v)]==_ ~ [P++1]],
where the functions p and Tj, introduced in l9 1, are even
functions of I; and V is the voltage potential at the S-N
junction (the charge e in included in V). The rate of
change of n as a result of pair injection will be equal
to zero, a fact which can be directly verified by computing the corresponding source. To the right-hand
side of (5) must also be added the collision integral.
Thus, in the steady-state case we have
L[
p+ l.. 1]] =
2
e
n\;p~ J w'dw
2eD'
JdS'{ [II(E'-i'-W) tn' (1-n)
x (HNo) -n(l-n')Nw ]H(i'-E'-w) tn' (1-n)N.
__
~
-nO-n') (l+No)]] ( H
,
.. ,
(6)
~2
;\~,
-nn (l+N .. )]
) H(i'H'-w)[N.,(l-n) (i-n')
U'-t.'
(1-~
n,
where eD = ps, s is the velocity of sound, g is the
matrix element of the interaction with phonons, defined
by
2"\;",,
\",/(E)~~'--
n
JdT[u'n+v'(1-n)]= ~.; Jds [ ! n,
(7 )
where ~ is the nonequilibrium potential that was discussed above. It can be seen that the integral of the odd
part of n1 over E, (with the weight 1;/ E) determines the
potential ~, Le., ~ is determined by the asymmetry
with respect to 1; of the quasiparticle distribution function. Notice that if we compute the growth rate of ~
with the aid of (5) and (7), then the result coincides with
the formula (23) of[9], To find ~, let us multiply (6) by
1;/ f and integrate over 1;, Then
-E-W)
(8)
x (HN w ) ]-li(i'+i"-(o) [Nw (I-n) (I-n')-nn' (HN.)]}.
Linearizing (8), we find after simple transformations
that
Let us consider the junction S-N meas . Then from
the formula (5) of[9] we have
Ih·
Jds JdTe-dM~) [~R(T)G(-T)-Q(T)GA(_,)]
Jd~.!!.':2.
[G(I") -2 Lh ",-o,-<D GR(w)] =0 .
. 2"
21'
(10)
Let us represent G( w) in the form
G(w)=2ith
2~
ImGoRHG(",),
(11)
where G~( w) is the equilibrium retarded Green function and oG( w) is the nonequilibrium correction to
Go(w) due to the injection of quasiparticles. The expression for 1m G~ (w) has the form
(12 )
2
where u~ = 7'2(1 + E,/f) and v = 7'2(1 - E,/E). Substituting (11) and (12) into (10), we obtain
~ Jds [Ill £+0, +th 0,-£] =
2
+
Jds .dw [2<D
2rr
21'
27'
IIllJ dsoG(t,t)
(13)
1m Go"(w)~th ~ -Zlh ~ 1m (0'0"(",) -Go R ( " , ) ) ] ,
0",
21'
21'
where G~ (w) coincides with the function G~ ( w) if we
make the substitution I; - E, + ~ in the latter.
In calculating 6'1> Tinkham[2] took only the first term
on the right-hand side of (13) into account, since he assumed that only the quasiparticle distribution function
changes during the injection and that ~ = 0. Indeed, if
we set ~ = 0, then
ImIiG(t,
(9 )
t)~(~Ie)/)(2n-l)=2(~!e)n"
and Tinkham's result follows from (13):
S
-2I 'J~ ds [£+0'
th-.21'- + t h0,-£
- - ] =2 Joo d~-[n,(S)-n,(-s)].
21'
"e
where
1131
£
w
Q(w)= 2th 27' ~R«(O).
We shall, however, be interested not in the function n1
itself, but in some integral of it. In fact; let us write
down the expression for the change oN in the total number of particles in S[ 10], a change which should be equal
to zero:
Jds (s/e)'1]=- n~~: Jd", ds d;' (;+;') ",' (t.'/i"') {Ii (i"
n
Let us now turn to the establishment of the relation
between the observable voltage potential and the potential ~. This relation can be found if the formula (5)
of[9], as applied to the junctions S-N meas and S-Smeas
of the measuring circuit, is used. Integration over ~
makes the left-hand side of this formula vanish, since
it is assumed that no current flows through the measuring junctions. Let us choose the gauge X= 0, i.e., let
us assume that ~ = cp in S. The potential of the electrode Nme as is then equal to 1f'1 + ~,where 1f'1 is the
potential difference between Sand Nmeas . As will be
shown, there arise between the superconductors Sand
Smeas a potential difference ~ and an order-parameter
phase difference oX,
no(f)= [e;/T+1]-'.
x tn' (I-n) (HN w ) -n(1-n')N w ]H(r-i" -w) tn' (1-n)N w -n(l-n')
PlnJ
' n"
Q' = , ds..2-
Here we have taken account of the fact that in the case
when the junction is with a normal metal ~ R( w) = -ill,
and we have used the relation[9]
We shall seek the solution to (6) in the form
where
an ( -,
6 )'--,
t., t l le)]
+ (~
- <D =0,
Ue
e
2e"
21'
(9')
ThuS, the steady-state value of Q. is determined by the
frequency LlQ' which vanishes in the normal metal. The
time TQ, introduced by Clarke and Tinkham, characterizes the time necessary for the establishment of equilibrium between the branches of the spectrum (the
branch-mixing time).
= v 1111
=no (€') + n1,
IIN=II
Ye"-L\'
+("+e')'(n,:+N.+,,,)- (e'-e)'(n'+N.,_,)O(e' -e)],
g'=2n'\;p,/pm ["], ,Y.. =(e',,/T_1)-', f=Y(S+Q»)'+t.' ['0],
the dimensionless constant I:ph being of the order of
unity.
Joo---==
de'
[0'(e-e ') ( e-e'
')"(1 -no'"
-n._,' )
(::lv' , e'
_00
SOy. Phys.·JETP. Vol. 42, No.6
A. F. Volkov and A. V. Zartsev
1131
As was pointed out in[9], however, the first term in (13)
is the change in the total number of particles in Sand
is therefore equal to zero. In fact, the voltage potential
~ 1 is due to the second term. Computing it with the aid
of (12), we find
_ J d [th 8.+8 + th 8.-8]
~
2T
2T
=
4~'<DJ~
,
de
(c'-~')'"
a th(El2T)
(14)
iJE
The expression (14) coincides with one found earlier
in[9l.
Let us consider the junction S -Smeas. Proceeding
in the same way as in the determination of the source
JS-S in[9], we can derive from the formula (5) of[9] the
following expression for the phase difference Ox for the
order parameter in Sand Smeas:
=
- J~~
[~"R(Ol)Fo'(-Ol)Hll2(Ol)Fo<'(-Ol)] sin6X
2Jt
d
Re Jds ~ {LR(Ol)G(Ol) -Q(Ol)GA(Ol)-~12R(Ol)F' (-Ol)
2Jt
(15)
Hl" (Ol)F A' (-Ol) },
where in the expression on the left-hand side of the
equality figure the equilibrium functions F 0, while in
the expression on the right figure the nonequilibrium
functions G and F. The integral on the left-hand side
is proportional to the Josephson current. Let us denote
it by IJ. Let us express the functions G and F figuring
in the expression on the right-hand side of (15) in terms
of 1m GR and 1m FR and take into account the fact that
the integral of the second term in the curly brackets
vanishes on account of the fact that this term is an odd
function of w. Carrying out the computations, we find
(16)
Thus, the asymmetry of the quasiparticle distribution
leads to the appearance in the system S-Smeas of a
current that is canceled by the pair current. As a result of this, the phase difference Ox arises and, furthermore, a potential difference exists between Sand
Smeas, since in computing (15) we assumed the potential
of Smeas to be equal to zero. The observable voltage
potential is made up of <I> and ~1: ~ = ~l + <1>.
Let us find the dependence ~(V, T) for T - Tc. It
follows from (14) that, near the critical temperature,
Iff 1 ~ (t. IT )2<1> « <1>, and therefore the observable voltage
potential is due to the potential difference between S
and Smeas. In the expression (9') for vQ( E) only the
first and second terms are important near Tc. Computing them, we obtain
(17)
The expression (17) coincides up to a numerical factor
with the expression found by Tinkham[2l. In the integral
(9), which contains IIQ( £), the characteristic scale of
the variation of n1 for T - Tc is the temperature;
therefore, vQ is determined by the expression (17) with
the energy E replaced by T*, where the "temperature"
T* is of the order of T and should be determined from
the exact solution to the kinetic equation (8). From Eq.
(7) we find the relation between <I> and Q*:
<D="';:"Q''''
pm
Jd~n.-.i=-4(v/vQ)V.
8
Substituting vQ from (17) into this expression, we
finally find
(T') (-eT·
8 = - 4\'
-- Jt'~Ph
1132
~
D )'
(T' ) V.
th 2T
Sov. Phys.-JETP, Vol. 42, No.6
The dependence of ~ on (T c - T) for t. « T agrees
qualitatively with the corresponding dependence observed experimentally in the case of Sn[ 3].
2. THE SYSTEM N\-N-N2
The system (Fig. 1) in question also allows us to investigate the mechanism responsible for energy relaxation in the case of a normal metal. Then instead of the
system S'-S-N, we should use the system N 1-N-N 2,
where N 1 and N2 may be of the same metal: It is only
necessary that the frequencies VI and 112, which are
proportional to the product of the density of states of
the metal N1 (N2) and the matrix element for tunneling
through the N 1-N (N 2-N) junction, differ from each
other. Then in the presence of a current flowing through
the system NcN-N2 the distribution function in N will
become asymmetric in ~,although, as follows from the
neutrality condition, the total number of electrons will
remain equal to the total number of holes. In the measuring circuit will then arise a potential difference Iff.
In fact, using Eq. (7) of[9] and the expression for the
self-energy part describing the tunneling from Smeas
into N,
L:"=-i\' I '" I [Ol'-I\']-'1.0 (I Oll-~),
we obtain for the voltage potential Iff in the Smeas-N
circuit the equation
J d~(th
8+8 -th
2T
E-8)=_2Jd~ ~n,(~) O(I~I-~)
2T
(~'_~') '1.
(18)
'
where E = (~~ + t.~)l/2 and n1 is the deviation of the
distribution function in N from the equilibrium distribution function 2). It can be seen from (18) that if the
measuring electrode is a normal metal, then the integral on the right -hand side (and, consequently, ~) will
vanish because of the electrical-neutrality condition.
Let us find the quantity n1' Let a current flow through
the system, so that we have established at the N eN and
N 2-N junctions the voltage potentials VIand V 2 respectively. The kinetic equation for n1 has the form
(19 )
where the second term on the right-hand side is the
source due to the injection of quasi particles from N 1
and N2, the frequencies Ilk are connected with the resistances of the junctions Nk-N[9], and 1st is the
linearized collision integral for collisions with phonons.
We shall restrict ourselves to the case of low temperatures (T« t.) and shall assume that Vk < ® D. After
integration over the angles Eq. (19) in the steady-state
case assumes the form
1
~£3n.(~)-8(S)
=
2e D '
n~Ph
I
I
,
~
dOl(Ol-s)'n,(Ol)+8(-S)
dOl (0l-s)2 n ,(0l)
[\',O(~)8(IV,I-s)-\,0(-s)e(S+V2)],
(20)
where VI < 0 and V 2> O. We seek the solution in the
form
I
nl(~) =n>O(~) O( VII-~) +n<O (-s)O(~+ V 2 ) .
(21)
Then for the electron distribution function we obtain the
equation
1
~
W~
_~3n>_ dOl(0l-~)2n>(0l)=xl"'-.-v"
J
3,
s>o.
(22)
n,ph
For the hole distribution function we obtain an equation
A. F. Volkov and A. V. Za'tseY
1132
coinciding with (22) if we make the substitutions
~ - -~, VI - VOl, and Kl - K2 in the latter, Differentiating (22) three times with respect to ~. we obtain
(23)
1 iP(s'n»
----+2n>~O.
:T(a,Y>
15
12 -
os'
3
The solution to (23) is the power function
n>£'~C.s-'+ Rc (C,+iC,) :;""'.
Let us determine the constants from the boundary conditions
-J
which follow from Eq. (22) and its derivatives. The
solution then assumes the form 3)
n
>
~~-=-{(~)'11 Iv.I'
where sincpo
l'33
S
()
~sin(l'2In-s -S
IV.I
<p
o)},
(24)
= 5/ m.
Let us substitute the solutions n > (~, Kl) and
n«~, K2) = n>(-;, K2) into (21) and nl from (21) into
(18). Then, assuming that E < T, we obtain
0~
l_(~)'I'eA!T{Sv'dSsn>(s,",) _lsV>ld~sn>(s,",)
2l'2rt
L1
(£'-L1')'/'
A
(5'-L1')"
A
(25)
where
l'a'-l
"
1
£T(a, 1) = - 2- -l' (aly)'-l--:;- + arccos2~
a
- arccos l
-
a
Ja 2
-:-s
l'33
+-.
Jcc
ex
}33
x
-.-S
dx(x'-1)-" sin ("I'2In--'I'0)
e
2
-
a
1
en
_
X
dx(x2-1)-·"Sin(l'2In---.l-<po),
1
a
a = \V 1\ / A, Y = K2/ Kl = Rl /R2 , and Rl ,2 are the resistances of the junctions N 1,2-N. In this case in deri ving
(25) we used the neutrality condition IINl = IIN2, which
follows from Eq. (19) if we integrate this equation (in
the steady-state case) over ~.
Let us give the asymptotic expressions for §(a, y),
in the case when y » 1:
£T(a, 1) =2V~ npll a-+l,
£T(a, 1)=a[rrl2--"/;V(a!j)'-l) IIpl! a-+l·
The form of the function
(a, y), obtained by a numerical integration, is shown in Fig. 2 for several values of
y. Let us estimate the magnitude of the effect, setting
®D/A ~ 10 2, III ~ 10 6 sec- l [9 1, and §(OI, y) ~ 1. We obtain C ~ lO,jT/Ae A / T MV. Thus, by measuring the
function C (V 1) we can make judgments about the distribution function of the nonequilibrium electrons and
about the mechanism responsible for its relaxation.
Allowance for the electron-electron colliSions leads to
the appearance in (20) of terms of the type (~~/€F)nl'
These terms can be neglected provided A » ®O!E:F'
The flopover processes, which are neglected by us,
will be unimportant if the Fermi surface does not get
close to the Brillouin-zone boundaries.
3. CONCLUSION
Thus. the above-considered system allows us to investigate the asymmetry of the populations of the
energy-spectrum branches of a superconductor and, in
particular, determine the important characteristic TQ,
1133
SOy. Phys.-JETP, Vol. 42, No.6
FIG. 2. Dependence of F (a, ')') on a for,), = 2, 5, and 10.
the time for the establishment of equilibrium between
the branches. This time determines the attenuation
length of the longitudinal electric field in the superconductor. Notice that the gap aA also changes during the
tunnel injection. The change in the gap is due to the
first term on the left-hand side of (6). The time characterizing the establishment of the steady-state value of
A is less (near T cl than TQ and coincides in order of
magnitude with the energy-relaxation time in the normal
metal [131. The change in A does not (so long as it is
small) affect the magnitude of <1>, since liA does not
enter into Eq. (7).
The experimental investigation of the characteristic
function C(VI. T) in the case of the system N1-N-N z is
also of interest, since such a dependence allows us to
draw some conclusions about the mechanism responsible for energy relaxation in normal metals,
liThe potential 1> also arises near the core of a moving vortex [7,8].
2lThe quantity n l is an electron distribution function for ~ > 0 and a
hole distribution function for ~ < O.
3lThis solution diverges at small ~. To obtain a finite n> at ~ = 0, we
may take into account either the induced transitions or the nonlinear
terms in 1st and in the generation terms in (20). We are, however, not
interested in the region of small t since the contribution to (18) is
made by ~ ;;. A. Notice that in the case of the model matrix element
g2 _ q-2 of the interaction with the phonons, it is possible to solve
exactly the nonlinear kinetic equation with allowance for recombination of nonequilibrium electrons and holes. It then turns out that
the exact distribution function differs from the approximately determined function at energies ~:;:; (V IV,)1I2 In the case of the matrix
element used by us the nonlinear effects are important at ~
:;:; (Elb v IVI)1I4
J. Clarke, Phys. Rev. Lett. 28, 1363 (1972); M.
Tinkham and J, Clarke, Phys. Rev. Lett. 28, 1366
(1972 ).
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I
A. F. Volkov and A. V. Zartsev
1133
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1134
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Translated by A. K. Agyei
240
A. F. Volkov and A. V. Zartsev
1134